The smoothness indicators based on Lagrange interpolation polynomials
are derived in [1]} which gives the desired order
of convergence if the first and second derivatives vanish but the
third derivatives are non-zero by constructing the higher-order global
smoothness indicator in \(L_{2}-\) sense. The resulting scheme shows
less dissipative nature than other schemes and subsequently high order
schemes were presented in [2]}. Modified smoothness indicators
of WENO-JS scheme is presented in [3]} based on the linear
combination of second order derivatives over the global stencil by
using the idea of WENO-Z scheme. Acker et al. in [4]}, have
observed that the improving the weights information where the solution
is non-smooth is more important than the improving the accuracy at
critical points. To do this, they introduced an extra term to the
WENO-Z weights so that the scheme behaves with the same stability
and sharpness as the WENO-Z scheme at discontinuities and shocks with
a higher numerical resolution. The WENO methodology is still in development
to improve its rate of convergence in smooth regions and decrease
the dissipation near the discontinuities even it is successful in
a wide number of applications.
